Complex systems are dynamical systems consisting of a great many interacting parts. Typically we think of the parts as being agents with different degrees of intelligence and communicaton ability. In contrast, an ideal gas consists of a great many parts, the large number of molecules, that interact (through collisions). But the molecules themselves don't have much internal "intelligence", information processing capability, or computational capacity.

More typically the complex systems one has in mind have more sophisticated parts. Typical examples would be

An economy of competing firms

Ant species that develop sophisticated colonies and social structures

The brain's visual cortex

Schooling fish or flocking birds

Complex systems like these exhibit a number of charactistics, such as

Pattern formation in which the patterns take on functional utility

Spontaneous self-organization

The emergence of cooperation

Hierarchical structure

Collective properties beyond those directly contained in the parts

Very often, these complex systems are adaptive. Taking in external influences, they change their internal structure to take advantage of the new circumstances.